1. Riley, Ricardo and Reuben each fired six shots, and each got 71 points.
2. Riley's first two shots scored 22 points and Ricardo’s first shot scored only 3 points.
3. The scores for the 18 shots were as follows:

                 50-----1
                 25-----2
                 20-----3
                 10-----3
                 05-----2
                 03-----2
                 02-----2
                 01-----3

We observe that whoever scored the bullseye must also have scored a 3, since the only way to make 21 out of the remaining five shots, given the constraints in the problem, would be 10-5-3-2-1.

Riley obviously didn't score the bullseye.

Suppose Reuben scored a 3.  Then by symmetry, we'd have no way of determining whether Ricardo or Reuben scored the bullseye (and thus the puzzle wouldn't have unique a solution).

So we must assume that Reuben did not score a 3, and thus Ricardo scored the bullseye.